Properties

Label 16-12e24-1.1-c2e8-0-16
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 92·25-s + 4·49-s + 200·73-s + 760·97-s − 284·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.35e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 3.67·25-s + 4/49·49-s + 2.73·73-s + 7.83·97-s − 2.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 8·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(21.45703486\)
\(L(\frac12)\) \(\approx\) \(21.45703486\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
7 \( ( 1 - 2 T^{2} - 2397 T^{4} - 2 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 + 142 T^{2} + 5523 T^{4} + 142 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
17 \( ( 1 + p^{2} T^{2} )^{8} \)
19 \( ( 1 + p^{2} T^{2} )^{8} \)
23 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
29 \( ( 1 - 50 T + p^{2} T^{2} )^{4}( 1 + 50 T + p^{2} T^{2} )^{4} \)
31 \( ( 1 + 478 T^{2} - 695037 T^{4} + 478 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
41 \( ( 1 + p^{2} T^{2} )^{8} \)
43 \( ( 1 + p^{2} T^{2} )^{8} \)
47 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
53 \( ( 1 - 94 T + 6027 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 94 T + 6027 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59 \( ( 1 - 6862 T^{2} + p^{4} T^{4} )^{4} \)
61 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
67 \( ( 1 + p^{2} T^{2} )^{8} \)
71 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
73 \( ( 1 - 50 T - 2829 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 - 9118 T^{2} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 4178 T^{2} - 30002637 T^{4} - 4178 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 + p^{2} T^{2} )^{8} \)
97 \( ( 1 - 190 T + 26691 T^{2} - 190 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.79689671805417718512898171467, −3.35344150433852542322232328172, −3.35164476401223321120602136866, −3.33908590069716866549716745599, −3.27409818666700371604734190803, −3.26419613613599966692003804427, −2.96983775772443191495723507365, −2.75684779485349219653072775597, −2.74926773589679613739063475190, −2.61622407093589162073689523693, −2.39550814104313377710871444034, −2.29988315488131069726556744327, −2.04010529517000824752362310769, −2.02746237702728661581668228540, −1.89968797854150380301015951127, −1.83600833824568443042964103510, −1.48456698264684509174093350152, −1.36154866520004814764230080448, −1.11469235604203576448167252777, −0.987190930114747468350951789690, −0.922920546563537141627362439046, −0.63608244891083571944043796807, −0.61806009169192532007274278592, −0.35683936605372028514517666986, −0.27469055374492088567566089264, 0.27469055374492088567566089264, 0.35683936605372028514517666986, 0.61806009169192532007274278592, 0.63608244891083571944043796807, 0.922920546563537141627362439046, 0.987190930114747468350951789690, 1.11469235604203576448167252777, 1.36154866520004814764230080448, 1.48456698264684509174093350152, 1.83600833824568443042964103510, 1.89968797854150380301015951127, 2.02746237702728661581668228540, 2.04010529517000824752362310769, 2.29988315488131069726556744327, 2.39550814104313377710871444034, 2.61622407093589162073689523693, 2.74926773589679613739063475190, 2.75684779485349219653072775597, 2.96983775772443191495723507365, 3.26419613613599966692003804427, 3.27409818666700371604734190803, 3.33908590069716866549716745599, 3.35164476401223321120602136866, 3.35344150433852542322232328172, 3.79689671805417718512898171467

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.